### (0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

Rewrite Strategy: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

### (2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

The TRS R consists of the following rules:

h(f(x, y)) → f(y, f(h(h(x)), a))

Rewrite Strategy: INNERMOST

### (3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
f0(0, 0) → 0
a0() → 0
h0(0) → 1
h1(0) → 4
h1(4) → 3
a1() → 5
f1(3, 5) → 2
f1(0, 2) → 1
f1(0, 2) → 4
h2(0) → 8
h2(8) → 7
a2() → 9
f2(7, 9) → 6
f2(2, 6) → 3
f1(0, 2) → 8
f2(2, 6) → 7